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Omni-MATH / Find the largest positive integer k for which there exists a convex polyhedron P with 2022 edges, which satisfies the following properties: [list] [*]The degrees of the vertices of…

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problem

Find the largest positive integer kk{} for which there exists a convex polyhedron P\mathcal{P} with 2022 edges, which satisfies the following properties: [list] [*]The degrees of the vertices of P\mathcal{P} don’t differ by more than one, and [*]It is possible to colour the edges of P\mathcal{P} with kk{} colours such that for every colour cc{}, and every pair of vertices (v1,v2)(v_1, v_2) of P\mathcal{P}, there is a monochromatic path between v1v_1 and v2v_2 in the colour cc{}. [/list] [i]Viktor Simjanoski, Macedonia[/i]
Plain-text mathematical notation (without MathML)
Find the largest positive integer k for which there exists a convex polyhedron P with 2022 edges, which satisfies the following properties:
[list]
[*]The degrees of the vertices of P don’t differ by more than one, and
[*]It is possible to colour the edges of P with k colours such that for every colour c, and every pair of vertices (v₁,v₂) of P, there is a monochromatic path between v₁ and v₂ in the colour c.
[/list]
[i]Viktor Simjanoski, Macedonia[/i]
Original LaTeX notation
Find the largest positive integer $k{}$ for which there exists a convex polyhedron $\mathcal{P}$ with 2022 edges, which satisfies the following properties:
[list]
[*]The degrees of the vertices of $\mathcal{P}$ don’t differ by more than one, and
[*]It is possible to colour the edges of $\mathcal{P}$ with $k{}$ colours such that for every colour $c{}$, and every pair of vertices $(v_1, v_2)$ of $\mathcal{P}$, there is a monochromatic path between $v_1$ and $v_2$ in the colour $c{}$.
[/list]
[i]Viktor Simjanoski, Macedonia[/i]

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Source and history

task_id
837d56fb-81f5-56b5-a3b5-3ed83a1c95c4
task_key
test--837d56fb-81f5-56b5-a3b5-3ed83a1c95c4
task_revision_id
1
config / split
/ test

Official source

initial import